ARFIMA (long memory) models Christopher F Baum EC 327: Financial Econometrics Boston College, Spring 2013 Christopher F Baum (BC / DIW) ARFIMA (long memory) models Boston College, Spring 2013 1 / 44
ARFIMA (long memory) models ARFIMA (long memory) models In estimating an ARIMA model, the researcher chooses the integer order of differencing d to ensure that the resulting series ( 1 − L ) d y t is a stationary process. As unit root tests often lack the power to distinguish between a truly nonstationary ( I ( 1 ) ) series and a stationary series embodying a structural break or shift, time series are often first-differenced if they do not receive a clean bill of health from unit root testing. Many time series exhibit too much long-range dependence to be classified as I ( 0 ) but are not I ( 1 ) . The ARFIMA model is designed to represent these series. Christopher F Baum (BC / DIW) ARFIMA (long memory) models Boston College, Spring 2013 2 / 44
ARFIMA (long memory) models This problem is exacerbated by reliance on Dickey–Fuller style tests, including the improved Elliott–Rothenberg–Stock ( Econometrica , 1996, dfgls ) test, which have I ( 1 ) as the null hypothesis and I ( 0 ) as the alternative. For that reason, it is a good idea to also employ a test with the alternative null hypothesis of stationarity ( I ( 0 ) ) such as the Kwiatkowski–Phillips–Schmidt–Shin ( J. Econometrics , 1992, kpss ) test to see if its verdict agrees with that of the Dickey–Fuller style test. The KPSS test, with a null hypothesis of I ( 0 ) , is also useful in the context of the ARFIMA model we now consider. This model allows for the series to be fractionally integrated , generalizing the ARIMA model’s integer order of integration to allow the d parameter to take on fractional values, − 0 . 5 < d < 0 . 5. Christopher F Baum (BC / DIW) ARFIMA (long memory) models Boston College, Spring 2013 3 / 44
ARFIMA (long memory) models The concept of fractional integration is often referred to as defining a time series with long-range dependence , or long memory . Any pure ARIMA stationary time series can be considered a short memory series. An AR ( p ) model has infinite memory, as all past values of ε t are embedded in y t , but the effect of past values of the disturbance process follows a geometric lag, damping off to near-zero values quickly. A MA ( q ) model has a memory of exactly q periods, so that the effect of the moving average component quickly dies off. Christopher F Baum (BC / DIW) ARFIMA (long memory) models Boston College, Spring 2013 4 / 44
ARFIMA (long memory) models The ARFIMA model The ARFIMA model 1 The model of an autoregressive fractionally integrated moving average process of a timeseries of order ( p , d , q ) , denoted by ARFIMA ( p , d , q ) , with mean µ , may be written using operator notation as Φ( L )( 1 − L ) d ( y t − µ ) = Θ( L ) ǫ t , ǫ t ∼ i . i . d . ( 0 , σ 2 ǫ ) where L is the backward-shift operator, Φ( L ) = 1 - φ 1 L - .. - φ p L p , Θ( L ) = 1 + ϑ 1 L + ... + ϑ q L q , and ( 1 − L ) d is the fractional differencing operator defined by ∞ Γ( k − d ) L k ( 1 − L ) d = � Γ( − d )Γ( k + 1 ) k = 0 with Γ ( · ) denoting the gamma (generalized factorial) function. The parameter d is allowed to assume any real value. 1 See Baum and Wiggins ( Stata Tech.Bull. , 2000). Christopher F Baum (BC / DIW) ARFIMA (long memory) models Boston College, Spring 2013 5 / 44
ARFIMA (long memory) models The ARFIMA model The arbitrary restriction of d to integer values gives rise to the standard autoregressive integrated moving average (ARIMA) model. The stochastic process y t is both stationary and invertible if all roots of Φ( L ) and Θ( L ) lie outside the unit circle and | d | < 0 . 5. The process is nonstationary for d ≥ 0 . 5, as it possesses infinite variance; see Granger and Joyeux ( JTSA , 1980). Christopher F Baum (BC / DIW) ARFIMA (long memory) models Boston College, Spring 2013 6 / 44
ARFIMA (long memory) models The ARFIMA model Assuming that d ∈ [ 0 , 0 . 5 ) , Hosking ( Biometrika , 1981) showed that the autocorrelation function, ρ ( · ) , of an ARFIMA process is proportional to k 2 d − 1 as k → ∞ . Consequently, the autocorrelations of the ARFIMA process decay hyperbolically to zero as k → ∞ in contrast to the faster, geometric decay of a stationary ARMA process. For d ∈ ( 0 , 0 . 5 ) , � n j = − n | ρ ( j ) | diverges as n → ∞ , and the ARFIMA process is said to exhibit long memory, or long-range positive dependence. The process is said to exhibit intermediate memory (anti-persistence), or long-range negative dependence, for d ∈ ( − 0 . 5 , 0 ) . Christopher F Baum (BC / DIW) ARFIMA (long memory) models Boston College, Spring 2013 7 / 44
ARFIMA (long memory) models The ARFIMA model The process exhibits short memory for d = 0, corresponding to stationary and invertible ARMA modeling. For d ∈ [ 0 . 5 , 1 ) the process is mean reverting, even though it is not covariance stationary, as there is no long-run impact of an innovation on future values of the process. If a series exhibits long memory, it is neither stationary ( I ( 0 ) ) nor is it a unit root ( I ( 1 ) ) process; it is an I ( d ) process, with d a real number. A series exhibiting long memory, or persistence, has an autocorrelation function that damps hyperbolically, more slowly than the geometric damping exhibited by “short memory” (ARMA) processes. Thus, it may be predictable at long horizons. An excellent survey of long memory models—which originated in hydrology, and have been widely applied in economics and finance–is given by Baillie ( J. Econometrics , 1996). Christopher F Baum (BC / DIW) ARFIMA (long memory) models Boston College, Spring 2013 8 / 44
ARFIMA (long memory) models Approaches to estimation of the ARFIMA model Approaches to estimation of the ARFIMA model There are two approaches to the estimation of an ARFIMA ( p , d , q ) model: exact maximum likelihood estimation, as proposed by Sowell (1992), and semiparametric approaches. Sowell’s approach requires specification of the p and q values, and estimation of the full ARFIMA model conditional on those choices. This involves the challenge of choosing an appropriate ARMA specification. We first describe semiparametric methods, in which we assume that the “short memory” or ARMA components of the timeseries are relatively unimportant, so that the long memory parameter d may be estimated without fully specifying the data generating process. Christopher F Baum (BC / DIW) ARFIMA (long memory) models Boston College, Spring 2013 9 / 44
ARFIMA (long memory) models Semiparametric estimators for I(d) series The Lo Modified Rescaled Range estimator 2 The Stata routine lomodrs performs Lo’s ( Econometrica , 1991) modified rescaled range (R/S, “range over standard deviation”) test for long range dependence of a time series. The classical R/S statistic, devised by Hurst (1951) and Mandelbrot ( AESM , 1972), is the range of the partial sums of deviations of a timeseries from its mean, rescaled by its standard deviation. For a sample of n values { x 1 , x 2 , . . . x n } , k k Q n = 1 � � � x j − ¯ � � x j − ¯ � − Min 1 ≤ k ≤ n Max 1 ≤ k ≤ n x n x n s n j = 1 j = 1 where s n is the maximum likelihood estimator of the standard deviation of x . 2 See Baum and Röom ( Stata Tech. Bull. , 2001). Christopher F Baum (BC / DIW) ARFIMA (long memory) models Boston College, Spring 2013 10 / 44
ARFIMA (long memory) models Semiparametric estimators for I(d) series The first bracketed term is the maximum of the partial sums of the first k deviations of x j from the full-sample mean, which is nonnegative. The second bracketed term is the corresponding minimum, which is nonpositive. The difference of these two quantities is thus nonnegative, so that Q n > 0 . Empirical studies have demonstrated that the R/S statistic has the ability to detect long-range dependence in the data. Christopher F Baum (BC / DIW) ARFIMA (long memory) models Boston College, Spring 2013 11 / 44
ARFIMA (long memory) models Semiparametric estimators for I(d) series Like many other estimators of long-range dependence, though, the R/S statistic has been shown to be excessively sensitive to “short-range dependence,” or short memory, features of the data. Lo (1991) shows that a sizable AR ( 1 ) component in the data generating process will seriously bias the R/S statistic. He modifies the R/S statistic to account for the effect of short-range dependence by applying a “Newey–West” correction (using a Bartlett window) to derive a consistent estimate of the long-range variance of the timeseries. Christopher F Baum (BC / DIW) ARFIMA (long memory) models Boston College, Spring 2013 12 / 44
ARFIMA (long memory) models Semiparametric estimators for I(d) series For maxlag > 0 , the denominator of the statistic is computed as the Newey–West estimate of the long run variance of the series. If maxlag is set to zero, the test performed is the classical Hurst–Mandelbrot rescaled-range statistic. Critical values for the test are taken from Lo, 1991, Table II. Inference from the modified R/S test for long range dependence is complementary to that derived from that of other tests for long memory, or fractional integration in a timeseries, such as kpss , gphudak , modlpr and roblpr . Christopher F Baum (BC / DIW) ARFIMA (long memory) models Boston College, Spring 2013 13 / 44
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